Calculus Examples

$y = (3 - x^{2}) (x^{3} - x + 1)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 3 - x^{2}$ and $g (x) = x^{3} - x + 1$ .

$(3 - x^{2}) \frac{d}{d x} [x^{3} - x + 1] + (x^{3} - x + 1) \frac{d}{d x} [3 - x^{2}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{3} - x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x] + \frac{d}{d x} [1]$ .

$(3 - x^{2}) (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x] + \frac{d}{d x} [1]) + (x^{3} - x + 1) \frac{d}{d x} [3 - x^{2}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$(3 - x^{2}) (3 x^{2} + \frac{d}{d x} [- x] + \frac{d}{d x} [1]) + (x^{3} - x + 1) \frac{d}{d x} [3 - x^{2}]$

Step 2.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$(3 - x^{2}) (3 x^{2} - \frac{d}{d x} [x] + \frac{d}{d x} [1]) + (x^{3} - x + 1) \frac{d}{d x} [3 - x^{2}]$

Step 2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$(3 - x^{2}) (3 x^{2} - 1 \cdot 1 + \frac{d}{d x} [1]) + (x^{3} - x + 1) \frac{d}{d x} [3 - x^{2}]$

Step 2.5

Multiply $- 1$ by $1$ .

$(3 - x^{2}) (3 x^{2} - 1 + \frac{d}{d x} [1]) + (x^{3} - x + 1) \frac{d}{d x} [3 - x^{2}]$

Step 2.6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$(3 - x^{2}) (3 x^{2} - 1 + 0) + (x^{3} - x + 1) \frac{d}{d x} [3 - x^{2}]$

Step 2.7

Add $3 x^{2} - 1$ and $0$ .

$(3 - x^{2}) (3 x^{2} - 1) + (x^{3} - x + 1) \frac{d}{d x} [3 - x^{2}]$

Step 2.8

By the Sum Rule, the derivative of $3 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- x^{2}]$ .

$(3 - x^{2}) (3 x^{2} - 1) + (x^{3} - x + 1) (\frac{d}{d x} [3] + \frac{d}{d x} [- x^{2}])$

Step 2.9

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$(3 - x^{2}) (3 x^{2} - 1) + (x^{3} - x + 1) (0 + \frac{d}{d x} [- x^{2}])$

Step 2.10

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

$(3 - x^{2}) (3 x^{2} - 1) + (x^{3} - x + 1) \frac{d}{d x} [- x^{2}]$

Step 2.11

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$(3 - x^{2}) (3 x^{2} - 1) + (x^{3} - x + 1) (- \frac{d}{d x} [x^{2}])$

Step 2.12

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$(3 - x^{2}) (3 x^{2} - 1) + (x^{3} - x + 1) (- (2 x))$

Step 2.13

Multiply $2$ by $- 1$ .

$(3 - x^{2}) (3 x^{2} - 1) + (x^{3} - x + 1) (- 2 x)$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$(3 - x^{2}) (3 x^{2} - 1) + x^{3} (- 2 x) - x (- 2 x) + 1 (- 2 x)$

Step 3.2

Apply the distributive property.

$(3 - x^{2}) (3 x^{2}) + (3 - x^{2}) \cdot - 1 + x^{3} (- 2 x) - x (- 2 x) + 1 (- 2 x)$

Step 3.3

Apply the distributive property.

$3 (3 x^{2}) - x^{2} (3 x^{2}) + (3 - x^{2}) \cdot - 1 + x^{3} (- 2 x) - x (- 2 x) + 1 (- 2 x)$

Step 3.4

Apply the distributive property.

$3 (3 x^{2}) - x^{2} (3 x^{2}) + 3 \cdot - 1 - x^{2} \cdot - 1 + x^{3} (- 2 x) - x (- 2 x) + 1 (- 2 x)$

Step 3.5

Combine terms.

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Step 3.5.1

Multiply $3$ by $3$ .

$9 x^{2} - x^{2} (3 x^{2}) + 3 \cdot - 1 - x^{2} \cdot - 1 + x^{3} (- 2 x) - x (- 2 x) + 1 (- 2 x)$

Step 3.5.2

Multiply $3$ by $- 1$ .

$9 x^{2} - 3 x^{2} x^{2} + 3 \cdot - 1 - x^{2} \cdot - 1 + x^{3} (- 2 x) - x (- 2 x) + 1 (- 2 x)$

Step 3.5.3

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 3.5.3.1

Move $x^{2}$ .

$9 x^{2} - 3 (x^{2} x^{2}) + 3 \cdot - 1 - x^{2} \cdot - 1 + x^{3} (- 2 x) - x (- 2 x) + 1 (- 2 x)$

Step 3.5.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$9 x^{2} - 3 x^{2 + 2} + 3 \cdot - 1 - x^{2} \cdot - 1 + x^{3} (- 2 x) - x (- 2 x) + 1 (- 2 x)$

Step 3.5.3.3

Add $2$ and $2$ .

$9 x^{2} - 3 x^{4} + 3 \cdot - 1 - x^{2} \cdot - 1 + x^{3} (- 2 x) - x (- 2 x) + 1 (- 2 x)$

Step 3.5.4

Multiply $3$ by $- 1$ .

$9 x^{2} - 3 x^{4} - 3 - x^{2} \cdot - 1 + x^{3} (- 2 x) - x (- 2 x) + 1 (- 2 x)$

Step 3.5.5

Multiply $- 1$ by $- 1$ .

$9 x^{2} - 3 x^{4} - 3 + 1 x^{2} + x^{3} (- 2 x) - x (- 2 x) + 1 (- 2 x)$

Step 3.5.6

Multiply $x^{2}$ by $1$ .

$9 x^{2} - 3 x^{4} - 3 + x^{2} + x^{3} (- 2 x) - x (- 2 x) + 1 (- 2 x)$

Step 3.5.7

Add $9 x^{2}$ and $x^{2}$ .

$10 x^{2} - 3 x^{4} - 3 + x^{3} (- 2 x) - x (- 2 x) + 1 (- 2 x)$

Step 3.5.8

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 3.5.8.1

Move $x$ .

$10 x^{2} - 3 x^{4} - 3 + x \cdot x^{3} \cdot - 2 - x (- 2 x) + 1 (- 2 x)$

Step 3.5.8.2

Multiply $x$ by $x^{3}$ .

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Step 3.5.8.2.1

Raise $x$ to the power of $1$ .

$10 x^{2} - 3 x^{4} - 3 + x^{1} x^{3} \cdot - 2 - x (- 2 x) + 1 (- 2 x)$

Step 3.5.8.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$10 x^{2} - 3 x^{4} - 3 + x^{1 + 3} \cdot - 2 - x (- 2 x) + 1 (- 2 x)$

Step 3.5.8.3

Add $1$ and $3$ .

$10 x^{2} - 3 x^{4} - 3 + x^{4} \cdot - 2 - x (- 2 x) + 1 (- 2 x)$

Step 3.5.9

Move $- 2$ to the left of $x^{4}$ .

$10 x^{2} - 3 x^{4} - 3 - 2 \cdot x^{4} - x (- 2 x) + 1 (- 2 x)$

Step 3.5.10

Multiply $- 2$ by $- 1$ .

$10 x^{2} - 3 x^{4} - 3 - 2 x^{4} + 2 x \cdot x + 1 (- 2 x)$

Step 3.5.11

Raise $x$ to the power of $1$ .

$10 x^{2} - 3 x^{4} - 3 - 2 x^{4} + 2 (x^{1} x) + 1 (- 2 x)$

Step 3.5.12

Raise $x$ to the power of $1$ .

$10 x^{2} - 3 x^{4} - 3 - 2 x^{4} + 2 (x^{1} x^{1}) + 1 (- 2 x)$

Step 3.5.13

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$10 x^{2} - 3 x^{4} - 3 - 2 x^{4} + 2 x^{1 + 1} + 1 (- 2 x)$

Step 3.5.14

Add $1$ and $1$ .

$10 x^{2} - 3 x^{4} - 3 - 2 x^{4} + 2 x^{2} + 1 (- 2 x)$

Step 3.5.15

Multiply $- 2$ by $1$ .

$10 x^{2} - 3 x^{4} - 3 - 2 x^{4} + 2 x^{2} - 2 x$

Step 3.5.16

Subtract $2 x^{4}$ from $- 3 x^{4}$ .

$10 x^{2} - 5 x^{4} - 3 + 2 x^{2} - 2 x$

Step 3.5.17

Add $10 x^{2}$ and $2 x^{2}$ .

$12 x^{2} - 5 x^{4} - 3 - 2 x$

Step 3.6

Reorder terms.

$- 5 x^{4} + 12 x^{2} - 2 x - 3$